Fixed point sets of maps homotopic to a given map

• Main Author: Soderlund Christina L
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2006/46052

<p/> <p>Let <inline-formula><graphic file="1687-1812-2006-46052-i1.gif"/></inline-formula> be a self-map of a compact, connected polyhedron and <inline-formula><graphic file="1687-1812-2006-46052-i2.gif"/></inline-formula> a closed subset. We examine necessary and sufficient conditions for realizing <inline-formula><graphic file="1687-1812-2006-46052-i3.gif"/></inline-formula> as the fixed point set of a map homotopic to <inline-formula><graphic file="1687-1812-2006-46052-i4.gif"/></inline-formula>. For the case where <inline-formula><graphic file="1687-1812-2006-46052-i5.gif"/></inline-formula> is a subpolyhedron, two necessary conditions were presented by Schirmer in 1990 and were proven sufficient under appropriate additional hypotheses. We will show that the same conditions remain sufficient when <inline-formula><graphic file="1687-1812-2006-46052-i6.gif"/></inline-formula> is only assumed to be a locally contractible subset of <inline-formula><graphic file="1687-1812-2006-46052-i7.gif"/></inline-formula>. The relative form of the realization problem has also been solved for <inline-formula><graphic file="1687-1812-2006-46052-i8.gif"/></inline-formula> a subpolyhedron of <inline-formula><graphic file="1687-1812-2006-46052-i9.gif"/></inline-formula>. We also extend these results to the case where <inline-formula><graphic file="1687-1812-2006-46052-i10.gif"/></inline-formula> is a locally contractible subset.</p>